One of the most idiosyncratic training examples ever given to me of effective tai sabaki at arbitrary distances as you are describing was this suggestion:

Try this the next time you go into a men's restroom (or ladies' for those so inclined) with the typical swing in, self-closing door. Usually, one pushes it open with the arm, or straightarm it with your weight behind the arm.

Try just walking into it completely upright, arms at your sides and pirouette in a tight tenkan-tenkai 360 degrees as you make contact -- 1) Notice how the door moves,and 2) Notice how little impact or disturbance you receive in doing so.

Now try on your favorite uke.

I'm just hoping that "your favorite uke" isn't standing at one of the urinals.